We adapt the proof of [1]} Lemma 5.7.21 to our situation.
We can assume without loss of generality that \(\inf _{z \in C} V(x^*,z)>0\)  (else the assertion is trivial). For \(\inf _{z \in C} V(x^*,z)>\delta >0\) , we define
\(V_C^\delta :=(\inf _{z \in C} V(x^*,z)-\delta )\wedge 1/\delta >0.\) 
